1. Field of the Invention
The present invention relates to an improved model and accompanying algorithm to simulate and analyze ink ejection from a piezoelectric print head. The improvements to the simulation model and algorithm include the development of a central difference scheme for the coupled level set projection method for two-phase flows, and also to the construction of multi-grid-compatible discrete projection operators that can be used in the central difference scheme. The simulation model and algorithm may be embodied in software and run on a computer, with the time-elapsed simulation viewed on an accompanying monitor.
2. Description of the Related Art
The above-identified related application discloses a coupled level set projection method on a quadrilateral grid. In that invention, the velocity components and level set are located at cell centers while the pressure is at grid points. The Navier-Stokes equations are first solved in each time step without (considering the condition of incompressibility. Then, the obtained velocity is “projected” into a space of a divergence-free field. A Godunov upwind scheme on quadrilateral grids is used to evaluate the convection terms in the Navier-Stokes equations and the level set convection equation. A Taylor's expansion in time and space is done to obtain the edge velocities and level sets. Since the extrapolation can be done from the left and right hand sides for a vertical cell edge and can be done from the upper and lower sides for a horizontal edge, there are two extrapolated values at each cell edge (see equations (50)-(52) of the related application). Then, the Godunov upwind procedure is employed to decide which extrapolated value to take (see equations (53) and (54) of the related application). In the Godunov upwind procedure, a local Riemann problem is actually solved. For a Newtonian fluid, the local Riemann problem reduces to the one-dimensional Burger's equation, which has very simple solutions as given by equations (53) and (54) of the related application. So, basically, the local velocity direction decides which extrapolated value to use for a Newtonian fluid. However, if the fluid is not a Newtonian fluid, the local Riemann problem may not have simple solutions, and the Godunov upwind procedure would be very difficult to program as a result. The advantage of this kind of upwind scheme for advection terms is lower grid viscosity and higher numerical resolution. Its disadvantage is the increased code complexity needed to do the Taylor's extrapolation in both time and space and Godunov's upwind procedure.